Conventional bubble function element is explained. FIG. 47 is a schematic showing a conventional two-dimensional bubble function element, and FIG. 48 is a schematic showing a conventional three-dimensional bubble function element. As shown in FIGS. 47 and 48, a triangular (tetrahedral) bubble function element is expressed by the following equation (1) in the isoparametric coordinate system [r, s] ({r, s, t}) by using 4 (5) nodes composed of 3 (4) points forming the triangle (tetrahedron) and a center of gravity point (refer to non-patent documents 1, 2, and 3).
                    [                  Equations          ⁢                                          ⁢          1                ]                                                                                  u            h                    |                      Ω            e                          =                                                            ∑                                  α                  =                  1                                                  N                  +                  1                                            ⁢                                                Φ                  α                                ⁢                                  u                  α                                                      +                                          ϕ                B                            ⁢                              u                B                                              =                                                    Φ                T                            ⁢              u                        =                                          u                T                            ⁢              Φ                                                          (        1        )                                                      Φ            α                    =                                    Ψ              α                        -                                          1                                  N                  +                  1                                            ⁢                              ϕ                B                                                    ,                  α          =                                    1              ⁢                                                          ⁢              …              ⁢                                                          ⁢              N                        +            1                                              (        2        )            
In equation (1), Φα and φB represent the shape functions of bubble function element, uα and uB represent the values of each vertex (analytical physical quantity) of a triangle (tetrahedron) and the value of the center of gravity point (analytical physical quantity), and N represents the number of spatial dimensions. Shape functions are expressed by the following equations (3) to (6) in vector-based description.
[Equations 2]
Two-DimensionsΦT=[Φ1Φ2Φ3φB]  (3)uT=[u1u2u3uB]  (4)
Three-DimensionsΦT=[Φ1Φ2Φ3Φ4φB]  (5)uT=[u1u2u3u4uB]  (6)
In equation (2), ψα represents the shape function of the linear element of two dimensions or three dimensions, and is expressed by the following equations (7) and (8).
[Equations 3]
Two-DimensionsΨ1=1−r−s, Ψ2=r, Ψ3=s  (7)
Three-DimensionsΨ1=1−r−s−t, Ψ2=r, Ψ3=s, Ψ4=t  (8)
Shape function φB is called a bubble function. The bubble function is defined for each element such that the value is zero on the boundary of elements and is one at the center of gravity point. In an unsteady problem, a finite element formula adopting a bubble function element for spatial discretization can be expressed as the following equation (9).
[Equations 4]M{dot over (u)}+F(u)=0  (9)
In equation (9), u represents an unknown analytical physical quantity (pollutant concentration, temperature, discharge, water depth, flow velocity, pressure, displacement, etc.), M is a mass matrix, and F(u) is a term collectively containing terms other than the temporal differential term. As temporal discretization of equation (9), a four-step solution based on Taylor expansion is expressed by the following equations (10) to (13) (refer to non-patent document 4).
                              [                      Equations            ⁢                                                  ⁢            5                    ]                ⁢                                  ⁢                              Four            ⁢                          -                        ⁢            step            ⁢                                                  ⁢            solution                    ⁢                                          <                      1            ⁢            st            ⁢                                                  ⁢            step                    >                                                                              u                      n            +                          1              /              4                                      =                                            u              n                        -                                          M                                  -                  1                                            ⁢                                                Δ                  ⁢                                                                          ⁢                  t                                4                            ⁢                              F                ⁡                                  (                                      u                    n                                    )                                                              ⁢                                          <                      2            ⁢            nd            ⁢                                                  ⁢            step                    >                                    (        10        )                                          u                      n            +                          2              /              4                                      =                                            u              n                        -                                          M                                  -                  1                                            ⁢                                                Δ                  ⁢                                                                          ⁢                  t                                3                            ⁢                              F                ⁡                                  (                                      u                                          n                      +                                              1                        /                        4                                                                              )                                                              ⁢                                          <                      3            ⁢            rd            ⁢                                                  ⁢            step                    >                                    (        11        )                                          u                      n            +                          3              /              4                                      =                              u            n                    -                                    M                              -                1                                      ⁢                                          Δ                ⁢                                                                  ⁢                t                            2                        ⁢                          F              ⁡                              (                                  u                                      n                    +                                          2                      /                      4                                                                      )                                                                        (        12        )                                <                  4          ⁢          th          ⁢                                          ⁢          step                >                                                                      u                      n            +            1                          =                              u            n                    -                                    M                              -                1                                      ⁢            Δ            ⁢                                                  ⁢                          tF              ⁡                              (                                  u                                      n                    +                                          3                      /                      4                                                                      )                                                                        (        13        )            
The superscript n in equations (10) to (13) represents a known analytical physical quantity at present time n, and n+1 represents an unknown analytical physical quantity at the time after an infinitesimal time Δt has elapsed from a given time n.
Non-patent document 1: D. N. Arnold, F. Brezzi and M. Fortin, “A Stable Finite Element for the Stokes Equations”, Calcolo, Vol. 21, 1984, pp. 337-pp. 344
Non-patent document 2: J. C. Simo, F. Armero and C. A. Taylor, “Stable and Time-Dissipative Finite Element Methods for the Incompressible Navier-Stokes Equations in Advection Dominated Flows”, International Journal for Numerical Methods in Engineering, Vol. 38, 1995, pp. 1475-pp. 1506
Non-patent document 3: Jun-ichi Matsumoto, “Two-Level Three-Level Finite Element Method for Incompressible Viscous Flow Analysis Based on Bubble Functions”, Journal of Applied Mechanics (Japan Society of Civil Engineers), Vol. 7, August 2004, pp. 339-pp. 346
Non-patent document 4: Katsunori Hatanaka, “Computational Study on Forward and Inverse Analyses of Incompressible Viscous Fluid Flow by Multistep Finite Element Method”, PhD Thesis, Chuo University, 1993